Problem: Evaluate the definite integral. $\int^{5}_{2}\left(\dfrac{3x+3}{x^2}\right)\,dx = $ Choose 1 answer: Choose 1 answer: (Choice A) A $-\dfrac{21}{10}$ (Choice B) B $\ln\left(\dfrac{15}2\right)-\dfrac{9}{10}$ (Choice C) C $3\ln\left(\dfrac52\right)+\dfrac{9}{10}$ (Choice D) D None of the above
Solution: First, simplify and use the power and natural log rules: $\begin{aligned}\int^{5}_{2}\left(\dfrac{3x+3}{x^2}\right)\,dx ~&=~\int^{5}_{2}\left(\dfrac{3x}{x^2}+\dfrac{3}{x^2}\right)\,dx \\&=~\int^{5}_{2}\left(\dfrac{3}{x}+3x^{-2}\right)\,dx \\&=\left(3\ln(x)-3x^{-1}\right)\Bigg|^{5}_{2}\end{aligned}$ Second, plug in the limits of integration: $(3\ln({5})-3\cdot{5}^{-1})-(3\ln({2})-3\cdot{2}^{-1}) = 3\ln\left(\dfrac52\right)+\dfrac{9}{10}$. The answer: $\int^{5}_{2}\left(\dfrac{3x+3}{x^2}\right)\,dx~=~3\ln\left(\dfrac52\right)+\dfrac{9}{10}$